A353226 Expansion of e.g.f. (1 - x^2)^(-x).

1, 0, 0, 6, 0, 60, 360, 1680, 20160, 151200, 1663200, 17962560, 219542400, 2854051200, 40441040640, 606356150400, 9793028044800, 166481476761600, 3017626733721600, 57359043873331200, 1153275200453376000, 24233844054131712000, 535361100608439705600

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+1)/2)} (2*k-1)/(k-1) * a(n-2*k+1)/(n-2*k+1)!.

a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(k,n-2*k)|/k!.

a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (2*exp(n)). - Vaclav Kotesovec, May 04 2022

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2)^(-x)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^2))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, (i+1)\2, (2*j-1)/(j-1)*v[i-2*j+2]/(i-2*j+1)!)); v;
(PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(k, n-2*k, 1))/k!);

Crossrefs

Cf. A066166, A121452, A351155, A353227.

Sequence in context: A305331 A169769 A357966 * A191688 A375588 A375561

Adjacent sequences: A353223 A353224 A353225 * A353227 A353228 A353229

Keyword

nonn

Author

Seiichi Manyama, May 01 2022

Status

approved